Problem 16 - Precalculus with Limits

Question

In Exercises 13-18, find the inclination \(\theta\) (in radians and degrees) of the line with a slope of \(m\). \(m = 2\)

Step-by-Step Solution

Answer

The inclination of the line with a slope of 2 is approximately 1.11 radians or 63.43 degrees.

Step 1: Identify the slope

From the exercise, the slope \(m = 2\).

Step 2: Calculate inclination in radians

Use the tangent inverse function to calculate \(\theta\) in radians. \(\theta_{radians} = \tan^{-1}(m) = \tan^{-1}(2)\).

Step 3: Convert radians to degrees

To convert the angle from radians to degrees, use the relation \(\theta_{degrees} = \theta_{radians} * (180 / \pi)\). Substituting the value of \(\theta_{radians}\) from the previous step into this relation gives the angle in degrees.

Step 4: Evaluate Formulas

Evaluate \(\theta_{radians} = \tan^{-1}(2)\) using a scientific calculator to get the angle in radians. Then, input the radians value into the formula \(\theta_{degrees} = \theta_{radians} * (180 / \pi)\) to get the angle in degrees.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness. It's a crucial concept in geometry and calculus. The slope, often represented as \(m\), describes how much the line rises or falls as it moves from left to right on a graph. The formula for calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

A positive slope means the line goes upwards, whereas a negative slope indicates a downward direction. If the slope is zero, the line is horizontal. In our exercise, the slope is given as \(m = 2\), indicating a moderately steep line.Understanding slope is foundational because it allows us to predict the direction and angle of a line, which leads us to the next concept - finding the angle or inclination of the line.

Tangent Inverse Function

The tangent inverse function, denoted as \(\tan^{-1}(x)\) or \(\arctan(x)\), is used to find the angle whose tangent value is \(x\). In the context of slope, we use the tangent inverse function to find the inclination of a line. When a line has a slope \(m\), its inclination \(\theta\) in radians can be found using the formula:

• \(\theta_{radians} = \tan^{-1}(m)\)

In our step-by-step solution, we calculated \(\theta_{radians}\) for a slope of 2 by finding \(\tan^{-1}(2)\). Understanding how to use the tangent inverse function is critical, as it opens the door to converting between different angle measurements, an essential skill in both trigonometry and calculus.

Radian to Degree Conversion

Angles can be measured in radians or degrees, two units that are commonly used in mathematics. Understanding how to convert between these units is vital in many areas ranging from geometry to engineering. Radians are often used in calculus because they provide a more natural measure of angles in terms of the unit circle.To convert an angle from radians to degrees, use the formula:

• \(\theta_{degrees} = \theta_{radians} \times \frac{180}{\pi}\)

This formula is derived from the fact that \(\pi\) radians are equal to 180 degrees. So, multiplying an angle in radians by \(\frac{180}{\pi}\) results in its equivalent in degrees.In our exercise, after finding \(\theta_{radians} = \tan^{-1}(2)\), we used this conversion method to calculate \(\theta_{degrees}\). Understanding this conversion process is key to switching between the two measurement systems smoothly.